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Palczewski, J and Schenk-Hoppe, KR (2010) Market selection of constant proportions investment strategies in continuous time. Journal of Mathematical Economics, 46 (2). 248 -266. ISSN 0304-4068

https://doi.org/10.1016/j.jmateco.2009.11.011

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Market Selection of Constant Proportions

Investment Strategies in Continuous Time∗

Jan Palczewski a

Klaus Reiner Schenk-Hoppe b

October 29, 2009

Abstract

This paper studies the wealth dynamics of investors holding self-financing portfolios in a continuous-time model of a financial market.Asset prices are endogenously determined by market clearing. Wederive results on the asymptotic dynamics of the wealth distributionand asset prices for constant proportions investment strategies. Thisstudy is the first step towards a theory of continuous-time asset pricingthat combines concepts from mathematical finance and economics bydrawing on evolutionary ideas.

JEL-Classification: G11, G12.Key words: evolutionary finance, wealth dynamics, endogenous asset prices,random dynamical systems.

1 Introduction

This paper aims at developing a theory of asset pricing that is based onthe market interaction of traders in a continuous-time mathematical financeframework. While mathematical finance has offered deep insights in thedynamics of portfolio payoffs under the assumption of an exogenous price

∗We are most grateful for comments by participants of the AMaMeF 2007 MidtermConference, Vienna, and the 2008 Hausdorff workshop on Finance, Stochastics and Insur-ance at the University of Bonn. Jesper Lund Pedersen provided helpful comments duringhis visit to the University of Leeds in 2007. Financial support by the Finance Market Fund,Norway (project Stability of Financial Markets: An Evolutionary Approach) and the Na-tional Center of Competence in Research “Financial Valuation and Risk Management,”Switzerland (project Behavioural and Evolutionary Finance) is gratefully acknowledged.

aSchool of Mathematics, University of Leeds, United Kingdom and Faculty of Mathe-matics, University of Warsaw, Poland. Email address: J.Palczewski@leeds.ac.uk

bLeeds University Business School and School of Mathematics, University of Leeds,United Kingdom. Email address: K.R.Schenk-Hoppe@leeds.ac.uk

1

process, economists prefer—in a market context—equating demand and sup-ply through an endogenous price mechanism. This market interaction ofinvestors plays a central role in our approach; it is modeled through theintroduction of endogenous prices (driven by demand and supply) in theclassical mathematical finance model. Randomness stems from exogenousasset payoff processes (dividends) and variation in traders’ behavior.

Our analysis focuses on the survival and extinction of investment strate-gies which is defined through the asymptotic outcome of the wealth dy-namics. This evolutionary view to financial markets is explored in discrete-time models by Evstigneev et al. [10, 11, 12] (see also their survey [9]). Inthe present paper these ideas are used to develop an evolutionary financemodel in continuous time. This approach incorporates market interaction oftraders into the workhorse model of mathematical finance. The continuous-time setting overcomes the problem of a priori setting a frequency of trade.It also defines a benchmark for the specification of discrete-time models inwhich the wealth dynamics is consistent over different trading frequencies(see Palczewski and Schenk-Hoppe [23]).

The wealth dynamics in our continuous-time evolutionary finance modelis described by a random dynamical system (Arnold [2]). This system canbe written as a non-linear differential equation with random coefficients thathas not been studied before. Its asymptotic analysis requires the applicationof techniques that are new in this context. Specifically, we use the conceptof a random fixed point (Schenk-Hoppe and Schmalfuss [28]), the ergodictheory for Markov processes (Anderson [1]) and the arcsine law for Markovchains (Freedman [14]).

Whilst the model is developed in the most general setting, our analy-sis is restricted to time-invariant investment strategies. These self-financingstrategies prescribe to rebalance a portfolio so as to maintain constant pro-portions over time. This class is quite common in financial theory and prac-tice, see e.g. Browne [7], Mulvey and Ziemba [21], Perold and Sharpe [26].The assumption of time-invariance considerably reduces the level of mathe-matical difficulty. This allows to place the main emphasis on the novel ideasdeveloped in this paper. Future research will aim to investigate the general,but mathematically more demanding, case of time-dependent strategies.

Our main result is the identification of a unique investment strategyλ∗ that is asymptotically optimal in a market in which only time-invariantstrategies are present. This strategy prescribes to divide wealth proportion-ally to the average relative dividend intensities of assets. We show thatany other time-invariant investment strategy interacting in the market willbecome extinct by losing its wealth to the strategy λ∗. This finding hasimplications for asset pricing. If at least one investor follows the strategyλ∗, asset prices converge to a ‘fundamental’ value which (except for risk-freebonds) does not coincide with the usual valuation because it is based onaveraging relative rather than absolute instantaneous dividend payments.

2

Our results provide the basis for a recommendation to portfolio managerstargeting the well-documented success of pairs-trading strategies, see e.g.Gatev et al. [15]. Investing according to the strategy λ∗ generates excessreturns (in the long-term) if there are assets whose relative valuation doesnot coincide with the benchmark. Indeed the proposed portfolio strategy isless risky than pairs-trading as it does not involve short positions.

The modeling approach presented here provides an alternative frame-work for portfolio optimization under the market impact of trades. Ratherthan postulating an exogenous price impact function for one large trader(e.g. Bank and Baum [3]), our model provides an endogenous mechanism forthe market impact of transactions (large and small). The impact increaseswith the size of a transaction but the precise market response depends onthe wealth distribution across investors and their investment strategies.

The evolutionary approach to asset pricing presented here is related toasset pricing theories based on the notion of excess returns, e.g. Luenberger[18] and Platen [24], which can be traced back to the setting of betting mar-kets studied by Kelly [17]. These theories, however, do not take into accountmarket interaction of traders. Stochastic general equilibrium models (whichdo have endogenous prices), on the other hand, suffer from the intercon-nectedness of consumption and investment decisions. This feature precludesclear-cut results on the long-term dynamics of asset prices if markets areincomplete, see e.g. Blume and Easley [6].

The paper is organized in the following fashion. Section 2 introduces themodel. Section 3 presents general selection results on the market dynamics.Section 4 considers the particular case of Markovian dividend intensities.Section 5 concludes. All proofs are collected in the Appendix.

2 The Model

This section derives the general evolutionary finance model in continuoustime.

Consider the following description of a financial market in continuoustime which is based on the standard approach in mathematical finance(e.g., Bjork [5] or Pliska [25]). There are K assets (stocks) with cumula-tive dividend payments D(t) = (D1(t), ...,DK (t)), t ≥ 0. Denote the priceprocess, which will be specified later, by S(t) = (S1(t), ..., SK(t)). Eachasset is in positive net supply of one. There are I investors. The port-folio (in numbers of physical units of assets) of investor i is denoted byθi(t) = (θi

1(t), ..., θiK(t)). His cumulative consumption process is given by

Ci(t). For a self-financing portfolio-consumption process (θi(t), Ci(t)), the

3

dynamics of investor i’s wealth V i(t) =∑K

k=1 θik(t)Sk(t) is

dV i(t) =K

∑

k=1

θik(t−)

(

dSk(t) + dDk(t))

− dCi(t) (1)

(t− denotes the left-hand limit). The self-financing property of the strategymeans that changes in value can be attributed either to changes in assetprices, dividend income or consumption expenditure. If there is a jumpin either price or dividend, the gain (or loss) is attributed to the investorholding the asset just prior to the occurrence of this event.

Each investor’s portfolio can be written as

θik(t) =

λik(t)V

i(t)

Sk(t)(2)

with some real-valued process λik(t), provided V i(t) 6= 0 and Sk(t) 6= 0.

The quantity λik(t) can be interpreted as the trader’s budget share allocated

to the holding in asset k. We will refer to λi(t) = (λi1(t), ..., λ

iK(t)) as an

investment strategy. It follows from the definition of V i(t) that λi1(t)+ . . .+

λiK(t) = 1. Since assets are in net supply of one, market clearing and (2)

implySk(t) = λ1

k(t)V 1(t) + . . . + λIk(t)V I(t) = 〈λk(t), V (t)〉, (3)

where λk(t) = (λ1k(t), ..., λ

Ik(t)), i.e. the market value of each asset is given

by the aggregate investment in that asset. Within a framework in which eco-nomic agents maximize utility, this pricing equation can be seen as a linearapproximation of the market demand function at an equilibrium. Equa-tion (3) gives an explicit description of the marginal impact of changes ininvestors’ strategies and wealth on the market-clearing price. The rela-tion of this pricing equation and demand functions is explained in detail inEvstigneev et al. [11].

Combining the self-financing condition (1) with (2) and (3), one obtains

dV i(t) =

K∑

k=1

λik(t−)V i(t−)

〈λk(t−), V (t−)〉

(

d〈λk(t), V (t)〉 + dDk(t))

− dCi(t) (4)

for i = 1, ..., I.The system of equations (4) provides the foundation of our approach.

It describes the wealth dynamics for given investment strategies. This ap-proach is opposite to the usual view in mathematical finance which derivesportfolio budget shares (investment strategies) from asset holdings (portfo-lios, see e.g. Pliska [25, Sect. 2.5 and 5.6] and Bjork [5, Sect. 6.2]). In a (verydifferent) economic context a similar approach is promoted by Shapley andShubik [29, Sect. III].

4

Suppose for each trader i = 1, ..., I a process λi(t) = (λi1(t), ..., λ

iK(t)) is

given with λik(t) > 0 for all k and λi

1(t) + . . . + λiK(t) = 1. Then (4) defines

the dynamics for the vector of investors’ wealth V (t) = (V 1(t), ..., V I(t))(provided it makes sense from a mathematical point of view). The maininnovation is that the traders’ strategies rather than the price process aretaken as a primitive of the model. The evolution of investors’ wealth dependson the investment strategies present in the market, the wealth distributionand the asset dividend payoffs. Asset prices, which are determined by theinvestment strategies and the wealth distribution, (3), are endogenous.

Our model is closed, i.e. all markets clear, because (3) ensures market-clearing for the financial assets, while Walras’ law gives market clearingfor the consumption good. To verify this claim, sum up equation (4) overi = 1, ..., I and denote C(t) =

∑Ii=1 Ci(t) and D(t) =

∑Kk=1 Dk(t) to obtain

dC(t) = dD(t) (5)

because∑K

k=1 λik(t) = 1 for all i, i.e. every trader exhausts his budget. The

price of the consumption good is normalized to one and, therefore, doesnot appear in (1). Asset prices are therefore relative prices, i.e. expressedin terms of the consumption good price. The consumption good does notplay the role of money because it cannot be stored and, thus, cannot beused for the intertemporal transfer of wealth. On the other hand, consols(financial assets with a constant payoff stream but with price risk) can beaccommodated in this model.

An alternative derivation of the above model is provided in Palczewskiand Schenk-Hoppe [23]. They obtain the dynamics (4) as the limit of thediscrete-time evolutionary finance model (Evstigneev et al. [10, 11, 12]) asthe length of the time-step tends to zero.

This paper studies the wealth dynamics (4) under the following assump-tions.

(A.1) The dividend dynamics is expressed by intensities, dDk(t) = δk(t)dtwith non-negative processes δk(t), k = 1, ...,K. Moreover, δ(t) =∑K

k=1 δk(t) > 0 is a continuous process of finite variation.

(A.2) Investment strategies are time-invariant, λik(t) = λi

k with λik > 0 and

∑Kk=1 λi

k = 1.

(A.3) The consumption rate is proportional to wealth, dCi(t) = c V i(t)dtwith some constant c > 0 which is independent of i.

Under these assumptions the wealth dynamics can be described by a sys-tem of differential equations with random coefficients. Solutions to such sys-tems have a sample-path-wise representation, which considerably facilitatesthe analysis. Relaxing these assumptions to allow for diffusion processes as a

5

description of both time-dependent strategies and the dividend process leadsto several complications; even the existence and uniqueness of solutions tothe wealth dynamics equation is an open problem. When restricting theset of time-dependent strategies to processes of finite variation and keepingassumption (A.1) in place still leads to the wealth dynamics that cannotbe described by a differential equation. Equation (4) has to be interpretedin the integral form and contains integrals with respect to time and λk(t),k = 1, . . . ,K. This implies that the long-term evolution of V (t) can be radi-cally changed by the short-term behavior of the investment strategies. Thismakes the analysis of the wealth dynamics much more involved and the re-sults will supposedly be less clear-cut than in the case studied in this paper.The analysis of the wealth dynamics under less restrictive assumptions thanthose imposed here is left for future research.

The assumption of time-invariance of investment strategies implies thatchanges in asset prices can be attributed to the wealth dynamics: condition(A.2) yields dSk(t) = d〈λk, V (t)〉 = 〈λk, dV (t)〉. Time-invariant strategiesyield portfolio positions with constant investment proportions over time.This class of strategies is quite common in financial theory and practice, seee.g. Browne [7], MacLean, Thorp and Ziemba [19], Mulvey and Ziemba [21],Perold and Sharpe [26]. The particular proportions (λi

1, ..., λiK) can depend

on a random event occurring at the initial time.Postulating a common consumption rate for all the investors ensures that

no trader has an advantage from a lower consumption rate (i.e. consumptionper unit owned). Consider the difference in the growth rate of the wealthof two investors i, j with identical strategies. Suppose investor i has a lowerconsumption rate, ci < cj . Then (4) implies

dV i(t)

V i(t−)−

dV j(t)

V j(t−)= (−ci + cj)dt > 0,

i.e. the growth rate of investor i’s wealth is always higher than that ofinvestor j. Assumption (A.3) therefore ensures a level playing-field for allthe investors. It also places the emphasis on the return-based ‘finance view’rather than the consumption-based ‘economics view’ in this line of research.

Under assumption (A.3) the total book value of the wealth V (t) =V 1(t) + . . . + V I(t) and the instantaneous total dividend δ(t) = δ1(t) +. . . + δK(t) are related by

V (t) = δ(t)/c (6)

(which is immediate from (5)). The relation (6) is merely a reformulation ofthe market-clearing condition for the consumption good. The total amountspent on consumption (measured in the price of the consumption good) isequal to its supply: cV (t) = δ(t). This imposes an additional condition onthe initial value (V 1(0), ..., V I(0)) because the market for the consumptiongood only clears if V (0) = δ(0)/c.

6

Under conditions (A.1)–(A.3) equation (4) can be written as

dV i(t) =

K∑

k=1

λikV

i(t−)

〈λk, V (t−)〉

(

〈λk, dV (t)〉 + δk(t)dt)

− cV i(t)dt (7)

with i = 1, ..., I.The following lemma shows that the dynamics (7) is well defined and

the solution to this integral equation generates a random dynamical systemwith continuous sample paths. The proof, as well as that of any other resultin this paper, is given in the Appendix.

Lemma 1 Assume (A.1)–(A.3). The dynamics (7) has a unique solutionfor every initial value V (0) ∈ R

I+ with V (0) = δ(0)/c. Moreover, the solution

V (t) has continuous sample paths of finite variation satisfying (6).

The dynamics for feasible initial values can be described by a one-dimensional system under the following assumption.

(A.4) There are two investors in the market, i.e. I = 2.

Define the relative wealth wi(t) = V i(t)/V (t), i = 1, 2. Since dwi(t) =[dV i(t) − wi(t)dV (t)]/V (t), (7) yields

dw1(t) = cw1(t)

∑Kk=1

λ1

k

[λ1

k−λ2

k]w1(t)+λ2

k

ρk(t) − 1

∑Kk=1

λ1

kλ2

k

[λ1

k−λ2

k]w1(t)+λ2

k

dt (8)

with ρk(t) = δk(t)/δ(t) denoting the relative dividend intensity. Models withjust two investors are common e.g. in evolutionary game theory where an‘incumbent’ competes against a ‘mutant.’

Lemma 2 Assume (A.1)–(A.4). Then one has:(i) The dynamics (8) is well-defined for every initial value w1(0) ∈ [0, 1].(ii) The solutions to (7) and (8) are related as follows:Suppose (V 1(t), V 2(t)) ≥ 0 is a solution to (7). If the initial value satis-

fies V (0) = δ(0)/c then w1(t) = V 1(t)/V (t) solves (8).Suppose w1(t) is a solution to (8). Then (V 1(t), V 2(t)) = (δ(t)/c)(w1(t),

1 − w1(t)) solves (7).

The dynamics (8) possesses the steady states, w1(t) = 0 and w1(t) =1. Each steady state corresponds to a situation in which one investmentstrategy holds the entire wealth. Therefore, the (relative) asset prices aregiven by that strategy. For instance, if w1(t) = 0,

Sk(t)

Sn(t)=

λ2k V 2(t)

λ2n V 2(t)

=λ2

k

λ2n

, k, n = 1, . . . ,K.

The remainder of the paper is concerned with the convergence of the wealthdynamics to the steady state w1(t) = 0.

7

3 Selection Dynamics

The main results on the long-term behavior of the wealth dynamics arepresented in this section. We identify a particular investment strategy which,under certain conditions, gathers all wealth regardless of the initial state.This constant proportions strategy is selected by the market through itsglobal dynamics.

Throughout the remainder of the paper we impose assumptions (A.1)–(A.4) and

(A.5) There are two assets, i.e. K = 2.

Under condition (A.5) each trading strategy can be represented by a real-valued random variable λi with λi

1 = λi and λi2 = 1 − λi. We further write

ρ(t) for ρ1(t) = δ1(t)/[δ1(t) + δ2(t)] (with ρ2(t) = 1 − ρ(t)).If both investors’ strategies are identical, λ1 = λ2, their relative wealth

is constant ((8) simplifies to dw1(t) = 0 dt) and the ratio of asset prices istime-invariant because (3) implies

S1(t)

S2(t)=

λ1V (t)

(1 − λ1)V (t)=

λ1

1 − λ1.

Assuming that λ1 6= λ2, (8) can be factorized as

dw1(t) = c−w1(t)

(

1 − w1(t))(

(λ2 − λ1)2w1(t) + (λ2 − λ1)(ρ(t) − λ2))

(

λ2(λ2 − 1) − λ1(λ1 − 1))

w1(t) + λ2(1 − λ2)dt.

(9)Our aim is to study the dynamics of (9) with the goal of finding a unique

investment strategy that is selected against any other strategy when bothinteract in the asset market.

Let us assume existence of the limit

ρ = limt→∞

1

t

∫ t

0ρ(s) ds (10)

with 0 < ρ < 1. In general the sample mean ρ is a random variable. Inmany applications, however, ρ is constant; for instance if ρ(t) is a positiverecurrent Markov process or a stationary ergodic process.

Define the time-invariant strategy

λ∗ = ρ. (11)

The construction of this investment strategy only uses ‘fundamental’ data:the investment proportions correspond to the time-average of the relativedividend intensities of the assets. This strategy has a game-theoretic inter-pretation, see Section 4.

The following theorem provides a result on the asymptotic dynamics ofa λ∗-investor’s relative wealth.

8

Theorem 1 Let λ2 = λ∗ and λ1 6= λ2. Then, for each initial value w1(0) ∈(0, 1),

(i) limt→∞

1

t

∫ t

0w1(s) ds = 0; and

(ii) limt→∞

1

t

∫ t

01[0,ε]

(

w1(s))

ds = 1 for all ε > 0.

The first assertion of Theorem 1 states that, for every sample path, therelative wealth of the λ∗-investor converges to 1 in the sense of Cesaro, whilethat of the other investor converges to 0. The second result shows that therelative wealth of the λ∗-investor asymptotically stays arbitrarily close to1. In this sense the market selects the λ∗-investor when competing with aninvestor who uses any other time-invariant strategy.

The convergence result in Theorem 1 cannot be strengthened withoutadditional assumptions on the relative dividend process. The asymmetryof the dynamics of w1(t) allows for the construction of a process ρ(t) suchthat w1(t) converges to 0 in the Cesaro sense but reaches a deterministiclevel l > 0 on every time interval [t,∞). The process w1(t) therefore doesnot converge to 0 in the usual sense. The estimates and results used in thefollowing example are derived in detail in the Appendix.

Counterexample Let λ1 = 1/4, λ2 = 1/2 and c = 1. The dynamics ofw1(t), (9), simplifies to

dw1(t) = −w1(t)(1 − w1(t))

(

w1(t) + 4(ρ(t) − 12)

)

4 − w1(t)dt. (12)

The following estimates hold: If ρ(u) ≤ 14 and w1(u) ≤ 1

2 for all u ∈ [s, t],0 ≤ s < t, then

w1(t) ≥ w1(s) exp(t − s

16

)

. (13)

If ρ(u) ≤ 12 and w1(u) ≤ 1

2 for all u ∈ [s, t], 0 ≤ s < t, then

w1(t) ≥1

27(t − s) + 1

w1(s)

. (14)

Fix any l ∈ (0, 1/2). If w1(0) ≥ l, let a0 = b0 = 0. Otherwise, let a0 = 0and b0 = 16 log

(

l/w1(0))

. Further, define for n ≥ 1

an =10n

l,

bn = 16n log(10),

9

and for t ≥ 0

ρ(t) =

12 if, for some k, t ∈

[

∑kn=0(an + bn),

∑kn=0(an + bn) + ak+1

)

,

14 if, for some k, t ∈

[

∑kn=0(an + bn) + ak+1,

∑k+1n=0(an + bn)

)

.

(15)

Lemma 3 The model with ρ(t) defined by (15) has the following properties:

(i) ρ = limt→∞

1

t

∫ t

0ρ(u)du =

1

2; and

(ii) w1(

∑kn=0(an + bn)

)

≥ l for all k ≥ 1.

The process ρ(t) fluctuates between the two values 1/4 and 1/2. Thelength of time during which it is equal to 1/4 grows linearly in n, whilethe time it is equal to 1/2 grows exponentially in n. This implies ρ(t) hasa long-term mean of 1/2 which is equal to λ2 and, therefore, investor 2follows the strategy λ∗. Lemma 3(i) and Theorem 1 ensure convergence ofw1(t) to 0 in the Cesaro mean. The asymmetry in the dynamics of w1(t),however, precludes that limt→∞ w1(t) = 0. If the relative dividend intensityρ(t) = 1/4, investor 1’s share of the total wealth increases exponentiallyfast. In contrast, if ρ(t) = 1/2, the relative wealth of investor 1 decreaseslinearly. This asymmetry holds as long as the relative wealth of investor 1is small enough, i.e. w1(t) ≤ 1/2. Lemma 3(ii) quantifies these findings: therelative wealth of investor 1, w1(t), reaches the level l > 0 at a sequence oftimes t converging to infinity, i.e. lim supt→∞ w1(t) ≥ l. �

The following Theorem presents conditions on the relative dividend in-tensity ensuring convergence of w1(t).

Theorem 2 Let λ2 = λ∗ and λ1 6= λ2. Fix a sample path of the dividendintensity. Suppose

(B) there is a real number z such that

lim supt→∞

1

t

∫ t

01(−∞,z)

(

sgn(ρ − λ1)

∫ t

s(ρ(u) − ρ) du

)

ds > 0.

Then, for every initial value w1(0) ∈ (0, 1), the relative wealth of investor 1converges to 0 (while the relative wealth of investor 2 converges to 1), i.e.

limt→∞

w1(t) = 0 and limt→∞

w2(t) = 1.

10

Theorem 2 strengthens the convergence result of Theorem 1 under a con-dition on the degree of asymmetry of ρ(t) with respect to its time-average ρ.The condition is trivially satisfied if ρ(t) is time-invariant, i.e. ρ(t) = ρ(0) forall t. The formulation of this result uses the sample-path wise interpretationof solution. If a realization of the process ρ(t) satisfies condition (B), thenconvergence w1(t) → 0 occurs for this particular sample path.

Under condition (B), if one investor divides the wealth in proportions(λ∗, 1−λ∗) while the other investor pursues a different time-invariant strat-egy, the first will overtake the latter: the relative wealth of the λ∗-investortends to one. The result holds globally, i.e. regardless of the initial wealthdistribution across the investors. Interpreting the dynamics as a fight be-tween an incumbent and a mutant strategy, the result says an incumbentpursuing the strategy λ∗ cannot be driven out of the market by any investorwith some other time-invariant strategy. In this sense, a λ∗-market is stable.

A benchmark for the asset prices can be derived from Theorem 2. Ifthe strategy λ∗ is present on the market, the asset prices have the followingasymptotic property:

limt→∞

S1(t)

S2(t)=

ρ

1 − ρ=

limt→∞1t

∫ t0 δ1(s)/(δ1(s) + δ2(s)) ds

limt→∞1t

∫ t0 δ2(s)/(δ1(s) + δ2(s)) ds

.

The ratio of the asset prices, in the long-term, is equal to that of the time-averages of the relative dividend intensities. This valuation is fundamentalas it only uses dividend data. For consols it coincides with the usual conceptof the fundamental value.

The portfolio positions of the λ∗-investor are given by

θ21(t)

θ22(t)

=λ∗

1 − λ∗

S2(t)

S1(t).

If the relative prices coincide with the benchmark (ρ, 1− ρ), the λ∗-investorholds the same number of units of each asset. Otherwise, he purchases aportfolio that is geared towards the undervalued asset. This asset alloca-tion implies that the λ∗-investor’s relative wealth grows asymptotically (atthe expense of the competitor). In this sense the λ∗-investor experiencesexcess growth even though the asset prices are endogenous and change theirstatistical behavior (which is non-stationary) over time.

The speed of convergence of w1(t) → 0 in Theorem 2 is not exponentiallyfast which is at odds with the corresponding models in discrete time, [10, 16].This observation follows from an analysis of the linearization at the steadystate w1(t) = 0. The variational equation at w1(t) = 0 of the dynamics (9)is given by

dv(t) = cλ1 − λ2

λ2(1 − λ2)

(

ρ(t) − λ2)

v(t) dt,

11

which shows that the exponential growth rate of v(t) is

cλ1 − λ2

λ2(1 − λ2)

(

ρ − λ2)

.

If λ2 = λ∗, the exponential growth rate is equal to zero for every investmentstrategy λ1. For any time-invariant investment strategy λ2 6= λ∗, however,there is an investment strategy λ1 such that the growth rate is strictlypositive, i.e. v(t) diverges from 0 exponentially fast. If λ2 < λ∗, take anyλ1 ∈ (λ2, 1); otherwise take λ1 ∈ (0, λ2).

4 Markovian Dividend Intensities

This section studies the condition imposed in the main convergence result,Theorem 2. The condition (B) is quite technical and does not reveal thetrue nature of the assumption imposed on the relative dividend intensityprocess ρ(t). The main obstacle is that, in general, different sample pathsof the process ρ(t) do not necessarily share properties beyond the requiredergodicity.

This section provides sufficient conditions for (B) under a particular as-sumption on the process ρ(t). The framework is that of time-homogeneousMarkov processes, which is at the heart of many models in financial mathe-matics and puts at our disposal a rich toolbox of results. The main findingis the identification of statistical regularities that are needed to obtain theconvergence result in Theorem 2.

Throughout this section the dividend intensity process ρ(t) is a posi-tively recurrent Markov process, i.e. the time of transition between any twostates is a finite random variable (which depends on a realization of the pro-cess). This assumption ensures that there are no transient states and thestate space contains one communicating set.1 For simplicity (and withoutrestriction on practical applications), the state space E of the process ρ(t)is assumed to be countable. We assume that E contains at least two states.Otherwise condition (B) is trivially satisfied.

The assumption of positive recurrence implies that there exists a uniqueinvariant probability measure µ for the process ρ(t). This measure has theproperty that µ({x}) > 0 for all x ∈ E. If µ is taken as the initial distribu-tion, the process ρ(t) is stationary. The interpretation of such a model is thatthe dividend-paying firms operate under stable, though random, conditions.

1Transient states (which are the states that are not visited after a finite time) canbe ignored because the analysis focuses on the long-run behavior of the market. Theassumption that there is one communicating set is not restrictive. If the state spaceconsists of several communicating sets, every such set can be considered separately, andthe investment strategy λ

∗ depends on the state of the dividend intensity process at time0.

12

Let Px denote the probability measure under which the distribution of

ρ(0) is concentrated in the state x. Pµ is the probability measure correspond-

ing to ρ(0) being distributed as µ. Since the state space E is countable, arelation between probability distributions P

x and Pµ has the following form:

Pµ =

∑

x∈E

µ(

{x})

Px. (16)

We collect some properties on the ergodicity of the positively recurrentprocess ρ(t) which will be needed in the following.

Lemma 4 If the initial distribution of ρ(t) is given by the invariant measureµ, then

Pµ(

limt→∞

1

t

∫ t

0ρ(s) ds = ρ

)

= 1,

whereρ = E

µρ(0)

and Eµ is the expectation with respect to measure P

µ. Moreover, if x ∈ E isthe initial state of ρ(t), then

Px(

limt→∞

1

t

∫ t

0ρ(s) ds = ρ

)

= 1.

This result ensures the time-invariant strategy λ∗ = ρ is well-defined andindependent of the initial distribution. If the dividend-paying firms operateunder stable economic conditions, which prevail when µ is the initial distri-bution, the strategy λ∗ states that one should invest according to expectedrelative dividend payments. It turns out that this strategy is selected bythe market dynamics even if the relative dividend process ρ(t) starts froman arbitrary state.

The strategy λ∗ has the following game-theoretic interpretation. For theincumbent’s constant strategy λ, the gross return on holding asset 1 overthe time [t, T ] is

Rt,T =λ + c

∫ Tt ρ(s)ds

λ

and, therefore, the instantaneous (marginal) return at time t is

∂Rt,T

∂T|T=t=

cρ(t)

λ.

Similarly, the instantaneous (marginal) return at time t on asset 2 is

c(1 − ρ(t))

1 − λ.

13

Define the concave function

λ 7→ arg maxη

Eµ log

(

cρ(t)

λη +

c(1 − ρ(t))

1 − λ(1 − η)

)

mapping an incumbent’s strategy to the set of “best responses,” i.e. all con-stant strategies maximizing the expected logarithmic growth rate at pricesgiven by the incumbent’s strategy λ. The first-order condition for a maxi-mum is

Eµ

[

ρ(t)λ − 1−ρ(t)

1−λρ(t)λ η + 1−ρ(t)

1−λ (1 − η)

]

= 0.

The incumbent’s strategy is the best response to itself (i.e. at the pricesdetermined by this strategy) if η = λ. In this case the above first-ordercondition is equivalent to

Eµ

[

ρ(t)

λ−

1 − ρ(t)

1 − λ

]

= 0,

which implies λ = Eµρ(t) = E

µρ(0) = λ∗. A comprehensive discussion ofthis line of results in evolutionary finance can be found in Evstigneev, Hensand Schenk-Hoppe [13].

We now turn to the main result of this section which relates the dy-namics of the sample mean of the process ρ(t) and the condition (B) ofTheorem 2. This theorem asserts the sample-path-wise convergence to 1 ofthe λ∗-investor’s relative wealth provided the other investor follows a differ-ent strategy.

Assume there exists a state x ∈ E such that if the process ρ(t) startsat x, the proportions of time its sample mean is strictly above, resp. below,the mean ρ are both positive:

(C)

limt→∞

1

t

∫ t

0P

x

(

1

s

∫ s

0ρ(u)du < ρ

)

ds > 0,

limt→∞

1

t

∫ t

0P

x

(

1

s

∫ s

0ρ(u)du > ρ

)

ds > 0.

The condition (C) implies the analogous condition for the process ρ(t) withthe initial distribution µ:

(C’)

limt→∞

1

t

∫ t

0P

µ

(

1

s

∫ s

0ρ(u)du < ρ

)

ds > 0,

limt→∞

1

t

∫ t

0P

µ

(

1

s

∫ s

0ρ(u)du > ρ

)

ds > 0.

Indeed, for every x ∈ E, the relation (16) and the fact that µ(

{x})

> 0imply the following inequalities:

Pµ

(

1

s

∫ s

0ρ(u)du < ρ

)

≥ µ(

{x})

Px

(

1

s

∫ s

0ρ(u)du < ρ

)

,

14

and

Pµ

(

1

s

∫ s

0ρ(u)du > ρ

)

≥ µ(

{x})

Px

(

1

s

∫ s

0ρ(u)du > ρ

)

.

Conditions (C) and (C’) are satisfied by a large class of processes. Ex-amples are presented below.

Theorem 3 Suppose the dividend intensity process ρ(t) is a positively re-current Markov process on a countable state space E satisfying assumption(C’). Then, for each initial state y ∈ E and for every real-valued ξ 6= 0,

lim supt→∞

1

t

∫ t

01(−∞,0)

(

ξ

∫ t

s(ρ(u) − ρ) du

)

ds > 0 Py − a.s.

Therefore, if λ2 = λ∗ and λ1 6= λ∗, then w1(t) → 0 (and w2(t) → 1) Py-a.s.

for every w1(0) ∈ (0, 1).

The condition in Theorem 3, which implies assumption (B), is writtenin statistical terms and, moreover, it is easily verifiable in applications.

Conditions (C) and (C’) exclude the asymmetry of the process ρ(t) thatwas exploited in the construction of the counterexample in Section 3. Theseconditions impose fluctuations of the sample mean of the relative dividendintensity process ρ(t) around its mean. In this realistic setting, the marketdynamics selects the λ∗-investor for almost every realization of the dividendprocess. This selection result holds independently of the initial distributionof the process ρ(t).

Two examples are discussed in the following.

Symmetric Markov processes. Assume that the process ρ(t) has initialdistribution µ and is symmetric around its expected value ρ = E

µρ(0), i.e.all finite-dimensional distributions of (ρ(t) − ρ) and (ρ − ρ(t)) under P

µ areidentical. Then, for every s ≥ 0,

Pµ

(∫ s

0

(

ρ(u) − ρ)

du < 0

)

= Pµ

(∫ s

0

(

ρ(u) − ρ)

du > 0

)

and, consequently,

Pµ

(∫ s

0

(

ρ(u) − ρ)

du < 0

)

=1

2

(

1 − Pµ

(∫ s

0

(

ρ(u) − ρ)

du = 0

))

.

Lemma 6, see the Appendix, implies

lims→∞

Pµ

(∫ s

0

(

ρ(u) − ρ)

du < 0

)

=1

2

and, by symmetry,

lims→∞

Pµ

(∫ s

0

(

ρ(u) − ρ)

du > 0

)

=1

2.

15

Thus condition (C’) is satisfied.

Exponentially ergodic Markov processes. Denote by τx the time ofthe first return of the process ρ(t) to the state x:

τx = inf{t ≥ σx : ρ(t) = x},

where σx is the first time of leaving x,

σx = inf{t ≥ 0 : ρ(t) 6= x}.

Both random variables are well-defined by right-continuity of the processρ(t).

Theorem 4 IfE

x(τx)2 < ∞

for some state x ∈ E, then condition (C) is satisfied.

The assumption of Theorem 4 ensures applicability of the Central LimitTheorem for Markov processes. While (by positive recurrence of ρ(t)) theexpectation E

xτx is finite for each initial state x, finiteness of the second mo-ment (variance) of τx is an additional condition. It is satisfied e.g. for expo-nentially ergodic Markov processes (as their transition probabilities convergeto the invariant distribution exponentially fast); cf. Anderson [1, Lemma6.3].

5 Conclusion

This paper develops a model of the market interaction of heterogenous in-vestment strategies within a continuous-time financial mathematics frame-work. The wealth dynamics is described by a random dynamical systemwhich is driven by an exogenous dividend process. We analyze the asymp-totic behavior of this dynamics for constant proportions investment strate-gies. Our results lead to the identification of a unique investment strategythat outcompetes any other time-invariant strategy. This finding has impli-cations for the long-term evolution of asset prices by providing an asymptoticbenchmark.

Future research on evolutionary finance models in continuous time willaim at the study of adapted, time-variant investment strategies, more in-vestors and more assets. Another line of inquiry is concerned with the corre-sponding diffusion-type model which requires the use of stochastic analysis.

16

Appendix

Proof of Lemma 1. Let us assume V i(0) > 0 for all i. The case of initialstates with one or more coordinates being equal to zero can be treatedanalogously as discussed later. Equation (6) yields

V I(t) =δ(t)

c−

I−1∑

i=1

V i(t) (17)

and, therefore,

dV I(t) =1

cdδ(t) −

I−1∑

i=1

dV i(t). (18)

Define the matrix Θ(t) ∈ R(I−1)×K by Θik(t) = λi

kVi(t)/〈λk, V (t)〉 with

V I(t) given by (17). Further, let the matrix Λ ∈ RK×(I−1) be given by

Λki = λik, k = 1, ...,K, i = 1, ..., I − 1, and define

A = Λ − ΛI ,

where ΛI ∈ RK×(I−1) has I − 1 identical columns, each being equal to the

vector λI . Then (7) is equivalent to the system of equations in V (t) :=(V 1(t), ..., V I−1(t)) which is given by

dV (t) = Θ(t−)(

AdV (t) +λI

cdδ(t) + δ(t)dt

)

− cV (t)dt

with δ(t) = (δ1(t), ..., δK (t))T . Equivalently, one can write

[Id − Θ(t−)A]dV (t) = Θ(t−)(λI

cdδ(t) + δ(t)dt

)

− cV (t)dt. (19)

The matrix [Id − Θ(t−)A] is invertible, which is shown at the end of thisproof. Equation (19) can be written in the explicit form

dV (t) = [Id − Θ(t−)A]−1[

Θ(t−)(λI

cdδ(t) + δ(t)dt

)

− cV (t)dt]

. (20)

Its canonical representation is

dV (t) = F(

δ(t), V (t−))

dZ(t), (21)

where Z(t) = (t, δ(t))T and

F (δ, v) = [Id − Θ(δ, v)A]−1(

Θ(δ, v)δ − cv, Θ(δ, v)λI

c

)

with

δ =K

∑

k=1

δk and Θik(δ, v) =λi

kvi

∑I−1i=1 (λi

k − λIk)v

i + λIk δ/c

. (22)

17

Define

D ={

(δ, v) ∈ [0,∞)K × (0,∞)I−1 :I−1∑

i=1

vi <1

c

K∑

k=1

δk

}

.

Continuous differentiability of F on D with respect to v implies F satisfiesthe following uniform local Lipschitz condition: for any (δ, v) ∈ D thereexist neighborhoods Oδ of δ and Ov of v such that

‖F (δ′, v′) − F (δ′, v′′)‖ ≤ K‖v′ − v′′‖, v′, v′′ ∈ Ov, δ′ ∈ Oδ

for some K > 0 depending on Oδ and Ov.A left-continuous version of equation (21) is given by

dV (t) = F(

δ(t−), V (t−))

dZ(t). (23)

Theorem 6 in [27, Chapter V] ensures that (23) has a unique local solutionfor every initial condition in (δ(0), V (0)) ∈ D. For every ω, the function t 7→δ(t)(ω) has finite variation. Continuity of F implies that F (δ(t), V (t−))(ω)can differ from F (δ(t−), V (t−))(ω) at most in a countable number of points.Continuity of Z(t) implies that the signed measure induced by the functionZ(t)(ω) is atomless, and, therefore,

∫ t

01F (δ(t−),V (t−))6=F (δ(t),V (t−))dZ(t) = 0.

The sets of solutions to equations (21) and (23) are identical, which ensureslocal existence and uniqueness of solutions to (21). The continuity of V (t)follows from the absence of atoms in the measure induced by the functionZ(t).

Since the set D is invariant, this solution is global, i.e. well-defined forall t ∈ [0,∞). Indeed, if V i(0) = 0 for one or more investors (but V j(0) > 0for at least one investor) then the absence of short-selling, (A.2), and (7)imply V i(t) = 0 for all t ≥ 0. The relation (6) together with assumption(A.1) ensures that a solution corresponding to a non-zero initial state cannotbecome identical to zero. The non-trivial (lower-dimensional) dynamics isdefined for all investors with strictly positive initial wealth. Existence anduniqueness of the solution follows from the above.

Proof of invertibility of [Id − Θ(t−)A]. Fix any (δ, v) ∈ D andlet Θ = Θ(δ, v), using the notation introduced in (22). We want to showinvertibility of [Id − Θ (Λ − ΛI)].

The matrix C = Id − Θ Λ has a column-dominant diagonal. Each diag-onal entry strictly dominates the sum of absolute values of the remainingentries in the corresponding column:

Cii >I−1∑

j=1,j 6=i

|Cji|, i = 1, . . . , I − 1. (24)

18

Indeed, the (i, j) entry of the matrix C is given by

1i=j −

K∑

k=1

Θikλkj .

Since all off-diagonal entries are negative, the condition (24) is equivalent to

I−1∑

i=1

K∑

k=1

Θikλkj < 1.

The following computation proves this inequality:

I−1∑

i=1

K∑

k=1

Θikλkj =

K∑

k=1

(

I−1∑

i=1

Θik

)

λkj <

K∑

k=1

λkj = 1.

One has∑I−1

i=1 Θik < 1 because the investment of investor I in asset k isstrictly positive (see assumption (A.2)) and assets are in net supply of 1.The above property of the matrix C implies that it is invertible and C−1

maps the non-negative orthant into itself (see [22, Corollary, p. 22 andTheorem 23, p. 24]).

Invertibility of [Id − Θ (Λ − ΛI)] is equivalent to invertibility of

[Id − Θ (Λ − ΛI)]C−1 = Id + Θ ΛIC−1.

It suffices to prove that x = 0 is the only solution to the linear equation

x = −ΘΛIC−1x. (25)

For any y ∈ RI−1, the particular form of the matrix ΛI implies ΘΛIy = by,

where

b =[

K∑

k=1

Θ1kλIk, . . . ,

K∑

k=1

Θ(I−1)kλIk

]Tand y =

I−1∑

i=1

yi.

The linear equation (25) therefore can only have solutions of the form x = αbwith α ∈ R. All coordinates of b are non-negative because short-sales are notallowed in the model. Assume that x = αb is a solution to (25), with α 6= 0and bi > 0 for at least one i = 1, ..., I − 1. The condition α 6= 0 implies thatb = −b C−1b, which further yields C−1b = −1. Since the matrix C−1 mapsthe non-negative orthant into itself and b ≥ 0, all coordinates of C−1b arenon-negative and C−1b =

∑I−1i=1 (C−1b)i ≥ 0—a contradiction. Hence, the

only solution to (25) is x = 0, which proves the invertibility of the matrix[Id − Θ(t−)A]. �

Proof of Lemma 2. Part (i). Let I = 2. Elementary calculations sufficeto derive (8) from the relation dw1(t) = [dV 1(t) − w1(t)dV (t)]/V (t) and

19

(7). Existence of a local solution follows as in Lemma 1. Note that alldenominators are strictly positive by assumption (A.2). w1(t) = 0 andw1(t) = 1 are fixed points. Therefore the solution is global.

Part (ii). Straightforward from the derivation of (8) from (7). �

Proof of Theorem 1. Fix a sample path of the dividend intensity process.Assertion (i): Equation (9) implies

(λ1(1 − λ1) − λ2(1 − λ2))w1(t) + λ2(1 − λ2)

w1(t)(1 − w1(t))dw1(t) (26)

= −c(

(λ2 − λ1)2w1(t) + (λ2 − λ1)(ρ(t) − λ2))

dt,

which is equivalent to

d log

(

w1(t)

(1 − w1(t))γ

)

= −c(λ2 − λ1)2w1(t) + (λ2 − λ1)(ρ(t) − λ2)

λ2(1 − λ2)dt (27)

where γ = 1 + (λ1(1 − λ1) − λ2(1− λ2))/(λ2(1 − λ2)) = λ1(1− λ1)/(λ2(1 −λ2)) > 0. Dividing the integral form of this equation by t, one obtains

1

tlog

(

w1(t)

(1 − w1(t))γ

)

−1

tlog

(

w1(0)

(1 − w1(0))γ

)

(28)

= −c(λ2 − λ1)2

λ2(1 − λ2)

1

t

∫ t

0w1(s) ds − c

λ2 − λ1

λ2(1 − λ2)

1

t

∫ t

0

(

ρ(s) − λ2)

ds.

Equation (10) implies that 1t

∫ t0

(

ρ(s) − λ2)

ds converges to zero. Since 0 <w1(t) < 1,

0 <1

t

∫ t

0w1(s) ds < 1.

Assume that lim supt→∞1t

∫ t0 w1(s) ds = 1 over the fixed trajectory of the

dividend intensity. There exists a sequence (sn) such that limn→∞ w1(sn) =1 and limn→∞

1sn

∫ sn

0 w1(s) ds = 1. (The construction of the sequence (sn)is provided at the end of the proof.) From equation (28) it follows that

limt→∞

γ

snlog

(

1 − w1(sn))

= c(λ2 − λ1)2

λ2(1 − λ2)> 0.

This contradicts log(

1 − w1(sn))

< 0 for all n.

Assume now that η = lim supt→∞1t

∫ t0 w1(s) ds is strictly positive, η > 0.

Clearly, η < 1. There exists a sequence (tn) (see below for its construction)such that

limn→∞

w1(tn) = η and limn→∞

1

tn

∫ tn

0w1(s) ds = η.

20

As n tends to infinity, the left-hand side of (28) converges to zero whereasthe right-hand side converges to −ηc(λ2 − λ1)2/(λ2(1 − λ2)) < 0. This is acontradiction.

The conclusion is that

lim supt→∞

1

t

∫ t

0w1(s) ds = 0.

Since w1(t) ≥ 0, this implies (i).Assertion (ii): Fix any ε > 0. Part (a) and w1(t) ≥ 0 yields

0 = limt→∞

1

t

∫ t

0w1(s) ds ≥ lim

t→∞

1

t

∫ t

01(ε,∞)(w

1(s))w1(s) ds

≥ limt→∞

ε

t

∫ t

01(ε,∞)(w

1(s)) ds ≥ 0.

Therefore, limt→∞1t

∫ t0 1(ε,∞)(w

1(s)) ds = 0, which implies (ii).Construction of the sequence (sn): Define

hn = inf{

t ≥ n : w1(t) ≥ 1 − 1/n}

,

sn = inf{

t ≥ hn :1

t

∫ t

0w1(s) ds ≥ 1 − 1/n

}

.

The sequences (hn) and (sn) are non-decreasing and converge to infinitybecause w1(t) < 1 and, by assumption, lim supt→∞

1t

∫ t0 w1(s) ds = 1. It

suffices to show that w1(sn) ≥ 1− 1/n to ensure that the sequence (sn) hasthe desired properties. If hn = sn, this relation holds. Otherwise one hassn > hn. Then there exists a z > 0 such that the function t 7→ 1

t

∫ t0 w1(s) ds

is non-decreasing for t ∈ [sn − z, sn]. This implies non-negativity of thederivative at t = sn:

w1(sn)

sn−

1

s2n

∫ sn

0w1(s) ds ≥ 0.

Hence, we have

w1(sn) ≥1

sn

∫ sn

0w1(s) ds = 1 − 1/n.

Construction of the sequence (tn): Define

h1n = inf

{

h ≥ n : sup{1

t

∫ t

0w1(s)ds : t ≥ h

}

≤ η +1

n

}

,

h2n = inf

{

t ≥ h1n : w1(t) ≥ η − 1/n

}

.

21

The sequences (h1n) and (h2

n) are non-decreasing and converge to infinitybecause η = lim supt→∞

1t

∫ t0 w1(s) ds. Let

tn = inf{

t ≥ h2n :

1

t

∫ t

0w1(s) ds ≥ η − 1/n

}

.

The sequence (tn) has the desired properties if

(a) (tn) is non-decreasing and converges to infinity,

(b) w1(tn) ≥ η − 1/n,

(c)1

tn

∫ tn

0w1(s) ds ≥ η − 1/n.

Part (a) and (c) follow directly from the definitions of h2n and tn, respectively.

If h2n = tn then (b) holds. Otherwise one has tn > h2

n. Then there existsa z > 0 such that the function t 7→ 1

t

∫ t0 w1(s) ds is non-decreasing on [tn −

z, tn]. Therefore, its derivative is non-negative at t = tn:

w1(tn)

tn−

1

t2n

∫ tn

0w1(s) ds ≥ 0.

This inequality, together with (c), implies (b). �

Proof of Lemma 3 and details of the counterexample. Fix λ1 = 1/4,λ2 = 1/2 and c = 1. The dynamics of w1(t) (see (12)) simplifies to

dw1(t) = −w1(t)(1 − w1(t))

(

w1(t) + 4(ρ(t) − 12)

)

4 − w1(t)dt. (29)

If ρ(u) ≤ 14 and w1(u) ≤ 1

2 for all u ∈ [s, t], 0 ≤ s < t, the followingestimate holds

w1(t) ≥ w1(s) exp( 1

16

∫ t

s

(

3 − 8ρ(u))

du)

≥ w1(s) exp(t − s

16

)

. (30)

Proof: Equation (29) is equivalent to

dw1(t) =w1(t)(1 − w1(t))

(

− w1(t) − 4(ρ(t) − 12)

)

4 − w1(t)dt.

If w(t) ≤ 12 and ρ(t) ≤ 1

4 , the right-hand side is nonnegative. An upperbound for the denominator is 4. A lower bound for the numerator is derivedusing the estimates (1 − w1(t)) ≤ 1

2 and

−w1(t) − 4(ρ(t) −1

2) ≤ −

1

2− 4(ρ(t) −

1

2) =

3

2− 4ρ(t).

22

Therefore,

dw1(t) ≥w1(t)

(

3 − 8ρ(t))

16dt. (31)

Integration of (31) yields (30). �

If ρ(u) ≤ 12 and w1(u) ≤ 1

2 for all u ∈ [s, t], 0 ≤ s < t, then

w1(t) ≥1

27(t − s) + 1

w1(s)

. (32)

Proof: If w(t) ≤ 12 , ρ(t) ≤ 1

2 , and w1(t) + 4(ρ(t) − 12) ≥ 0, then

dw1(t) ≥ −w1(t)

(

w1(t) + 4(ρ(t) − 12)

)

7/2dt ≥ −

2

7

(

w1(t))2

dt.

The above inequality is clearly satisfied if w1(t) + 4(ρ(t) − 12) < 0, because

this condition implies that the right-hand side of (29) is positive. Hence,the lower bound (32) for w1(t) is obtained by integration of the inequality

dw1(t) ≥ −27

(

w1(t))2

dt. �

The construction of the process ρ(t) is based on these two estimates. Fixany l ∈ (0, 1/2). If w1(s) ≥ l then w1(t) ≥ l10−n for

t ≤ s +10n

l2≤ s +

(10n

l− 1

) 7

2l

provided that ρ(u) ≤ 1/2 for u ∈ [s, t]. If w1(t) > 1/2 for some t ≥ s thenthe condition w1(t) ≥ l10−n is trivially satisfied. Otherwise, the estimate(32) implies the result.

The estimate for the case ρ(u) ≤ 1/4 is derived similarly. If w1(s) ≥l10−n then w1(t) ≥ l for

t ≥ s + 16n log(10)

provided that ρ(u) ≤ 1/4 for u ∈ [s, t]. If w1(t) > 1/2 for some t ≥ s thenthe condition w1(t) ≥ l is trivially satisfied. Otherwise, the estimate (30)implies the result.

The assertions of Lemma 3 follow straightforwardly from the above es-timates and the construction of the process ρ(t). �

Proof of Theorem 2. Define a new variable

v(t) =w1(t)

(1 − w1(t))γ, (33)

where γ = λ1(1 − λ1)/(λ2(1 − λ2)) as in the proof of Theorem 1. Transfor-mation (33) is a diffeomorphism between the interval (0, 1) of values of w1(t)

23

and (0,∞). Note that limt→∞ w1(t) = 0 if and only if limt→∞ v(t) = 0. Thelatter equality will be proved.

A differential equation for v(t) is obtained from (26):

dv(t) = −κ(t)v2(t) dt − q(t)v(t) dt,

where

κ(t) = c(λ2 − λ1)2

λ2(1 − λ2)(1 − w1(t))γ and q(t) = c

λ2 − λ1

λ2(1 − λ2)(ρ(t) − λ2).

The solution to this Ricatti equation is given by

v(t) =exp

(

∫ t0 q(s) ds

)

1/v(0) +∫ t0 κ(s) exp

(

∫ s0 qudu

)

ds.

This is not a closed-form expression for v(t) because the function κ(t) de-pends on w1(t). Since 1/v(0) > 0 one has

v(t) ≤1

∫ t0 κ(s) exp

(

−∫ ts q(u) du

)

ds.

Therefore limt→∞ v(t) = 0 if

limt→∞

∫ t

0κ(s) exp

(

−

∫ t

sq(u) du

)

ds = ∞.

Positivity of the integrand ensures that the above equality holds if and onlyif

lim supt→∞

∫ t

0κ(s) exp

(

−

∫ t

sq(u) du

)

ds = ∞.

For any ǫ > 0 and for any real number z the following sequence ofestimates holds

∫ t

0κ(s) exp

(

−

∫ t

sq(u) du

)

ds

≥ ǫe−z

∫ t

01[ǫ,∞)(κ(s)) 1(−∞,z)

(

∫ t

sq(u) du

)

ds

= ǫe−z

∫ t

0

(

1[ǫ,∞)(κ(s)) − 1[ǫ,∞)(κ(s))1[z,∞)

(

∫ t

sq(u) du

)

)

ds

≥ ǫe−z t

(

1

t

∫ t

01[ǫ,∞)(κ(s)) ds −

1

t

∫ t

01[z,∞)

(

∫ t

sq(u) du

)

ds

)

.

Choose

z =c|λ2 − λ1|

λ2(1 − λ2)z and any ǫ ∈

(

0,c(λ2 − λ1)2

λ2(1 − λ2)

)

.

24

Condition (B) is equivalent to

lim inft→∞

1

t

∫ t

01[z,∞)

(

∫ t

sq(u) du

)

ds < 1,

and Theorem 1 (b) implies that

limt→∞

1

t

∫ t

01[ǫ,∞)(κ(s)) ds = 1.

Therefore

lim supt→∞

(

1

t

∫ t

01[ǫ,∞)(κ(s)) ds −

1

t

∫ t

01[z,∞)

(

∫ t

sq(u) du

)

ds

)

> 0,

which implies lim supt→∞

∫ t0 κ(s) exp

(

−∫ ts q(u) du

)

ds = ∞. �

Proof of Lemma 4. The assertions of this lemma follow from the er-godic theorem for Markov processes (see Chapter II.14 in Chung [8]). Us-ing renewal theory, the lemma can also be proved employing Meyn andTweedie [20, Theorem 17.0.1] which is for discrete-time models. �

Proof of Theorem 3. Assume first that the distribution of ρ(0) is µ, i.e.the process ρ(t) is stationary. ρ(t) is defined for t ∈ [0,∞). Its stationarityallows the construction of a probability space and a stationary stochasticprocess ρ(t), t ∈ R, whose finite dimensional distributions for t ∈ [0,∞) areidentical those of ρ(t) (see Section A.3 in Arnold [2]). Define x(t) = ρ(−t).The process x(t) is a Markov process with the so-called adjoint semigroup(see Exercise 4.5.4, Stroock [30]). Assumption (C’) implies

limt→∞

1

t

∫ t

0P

µ

(

ξ

∫ s

0

(

x(u) − ρ)

du < 0

)

ds =: α ∈ (0, 1).

Freedman [14, Theorem 1]2 ensures that the family of random variables

1

t

∫ t

01(−∞,0)

(

ξ

∫ s

0

(

x(u) − ρ)

du)

ds

converges in distribution, as t → ∞, to a random variable with the cumula-tive distribution function

F (x) =

0, if x ≤ 0,sin(πα)

π

∫ x0 sα−1(1 − s)−αds, if x ∈ (0, 1),

1, if x ≥ 1.

2The paper studies only the discrete-time case. The regeneration approach employedthere generalizes to our case; see Freedman’s remark at the end of his paper [14].

25

Stationarity of x(t) ensures that the distribution of

1

t

∫ t

01(−∞,0)

(

ξ

∫ s

0

(

x(u) − ρ)

du)

ds

is identical to the distribution of

φ(t) :=1

t

∫ t

01(−∞,0)

(

ξ

∫ t

s

(

ρ(u) − ρ)

du)

ds.

The sequence of random variables φ(t) therefore converges in distribution toF (x) as t → ∞. Since F does not have an atom in 0, Lemma 5 (see below)implies that for any sequence tn converging to infinity

Pµ(

lim supn→∞

φ(tn) = 0)

= 0.

Using (16), one obtains for every y ∈ E

Py(

lim supn→∞

φ(tn) = 0)

≤1

µ(

{y})P

µ(

lim supn→∞

φ(tn) = 0)

= 0.

This finding implies

lim supt→∞

φ(t) > 0 Py − a.s.

which proves the first statement of the Theorem. Py-a.s. path-wise conver-

gence follows from this result and Theorem 2. �

Lemma 5 Let Xn be a sequence of random variables with values in [0,∞)such that Xn converges in distribution to X. If X does not have an atom at0, then

P(lim supn→∞

Xn = 0) = 0.

Proof. Let A = {ω : lim supn→∞ Xn(ω) = 0}. Since Xn ≥ 0, the set A canbe written as

A = {ω : limn→∞

Xn(ω) = 0}.

Define the function fǫ, for ǫ ∈ (0, 1), by

fǫ(x) = max(1 − x/ǫ, 0), x ≥ 0.

fǫ is continuous and bounded which implies Efǫ(Xn) → Efǫ(X). Let

Yn(ω) =

{

Xn, ω ∈ A,

1, ω /∈ A.

26

Obviously, Yn → 1Ac a.s. Since ǫ < 1, one further has

Efǫ(Xn) ≥ E(

1Afǫ(Xn))

= Efǫ(Yn).

The dominated convergence theorem gives

limn→∞

Efǫ(Yn) = Efǫ(1Ac) = P(A).

Hence,P(A) = lim

n→∞Efǫ(Yn) ≤ lim

n→∞Efǫ(Xn) = Efǫ(X).

Since X has no atom at zero, limǫ↓0 Efǫ(X) = 0. This implies P(A) = 0. �

Proof of Theorem 4. The proof is based on Central Limit Theorem forMarkov processes3 (see Theorem III.8.6 in Bhattacharya and Waymire [4]).Under the assumptions of the theorem

limt→∞

Px

(

(td)−1/2

∫ t

0

(

ρ(s) − ρ)

ds ≤ z

)

= Φ(z),

where Φ is a cumulative distribution function of the standard normal distri-bution, ρ = E

µρ(0), and d is a positive constant depending on x. Insertingz = 0 yields

limt→∞

Px

(∫ t

0

(

ρ(s) − ρ)

ds ≤ 0

)

=1

2.

Therefore,

limt→∞

1

t

∫ t

0P

x

(∫ s

0(ρ(u) − ρ)du ≤ 0

)

ds =1

2.

This clearly implies one assertion of (C):

limt→∞

1

t

∫ t

0P

x

(∫ s

0(ρ(u) − ρ)du > 0

)

ds =1

2.

The second part of (C) follows along the same lines; one simply has toreplace the process (ρ(t)) by (−ρ(t)). �

Auxiliary result. Consider the Markovian framework introduced in Sec-tion 4. Let ν be a distribution of ρ(0) (ν is the initial distribution of theprocess ρ(t)).

Lemma 6 One has

limt→∞

Pν

(∫ s

0

(

ρ(u) − ρ)

du = 0

)

= 0,

and, consequently,

limt→∞

1

t

∫ t

0P

ν

(∫ s

0

(

ρ(u) − ρ)

du = 0

)

ds = 0.

3One can also adapt Central Limit Theorem for discrete-time Markov chains, Meynand Tweedie [20, Theorem 17.2.2]. Its proof is based on a regeneration technique, whicheasily generalizes to the class of Markov processes considered in this paper.

27

Proof. The proof of the first assertion of the lemma is based on the followingidentity:

Pν

(∫ s

0

(

ρ(u) − ρ)

du = 0

)

= Pν

(

ρ(u) = ρ for all u ≤ s

)

. (34)

For its justification it suffices to show that the probability of

A =

{∫ s

0

(

ρ(u) − ρ)

du = 0, ρ(t) 6= ρ for some t ∈ [0, s]

}

is zero. This event can be decomposed into a countable number of events

An(ρ1, . . . , ρn) ={

there exists 0 = t0 < t1 < · · · < tn−1 ≤ tn = s :

ρ(u) = ρi for u ∈ (ti−1, ti),

n∑

i=1

(ρi − ρ)(ti − ti−1) = 0}

,

where n is a non-negative integer, ρ1, ..., ρn ∈ E, and ρi 6= ρi+1, i =1, ..., n − 1. The set An(ρ1, ..., ρn) represents the following event: there areexactly n−1 transitions of the process ρ(t) on the interval [0, s] through thestates ρ1, ..., ρn and the integral

∫ s0

(

ρ(u) − ρ)

du equals 0. To compute theprobability of An(ρ1, ..., ρn), the (n − 1)-dimensional density of transitionsthrough the states ρ1, ..., ρn is integrated on the set of (t0, ..., tn) such that

0 = t0 < t1 < · · · < tn−1 ≤ tn = s and

n∑

i=1

(ρi − ρ)(ti − ti−1) = 0.

This set has dimension n − 2, so the integral is equal to zero, and

Pν(

An(ρ1, . . . , ρn))

= 0.

Therefore, the probability of A is also zero because A is the union of acountable number of sets of measure zero. This completes the proof of (34).

Recall that the state space has at least two elements. If ρ is not an ele-ment of the state space, the first assertion of the lemma follows immediatelyfrom (34) because in this case P

ν(

ρ(u) = ρ ∀u ≤ s)

= 0. Otherwise, ifρ ∈ E, the probability that the process persists in the state ρ converges tozero,

lims→∞

Pν(

ρ(u) = ρ for all u ≤ s)

= 0.

This relation and (34) imply the first assertion of the lemma. The secondassertion of the lemma is a direct consequence of the first one. �

28

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